Theorem mulsrmo 8117

Description: There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Assertion

Ref	Expression
mulsrmo	⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))

Distinct variable groups:   𝑡,𝐴,𝑢,𝑣,𝑤,𝑧   𝑡,𝐵,𝑢,𝑣,𝑤,𝑧

Proof of Theorem mulsrmo

Dummy variables 𝑓 𝑔 ℎ 𝑞 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 8108	. . . . . . . . . . . . . . . 16 ⊢ ~R Er (P × P)
2	1	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ~R Er (P × P))
3		prsrlem1 8115	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))
4		mulcmpblnr 8114	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) → (((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔)) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩))
5	4	imp 124	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩)
6	3, 5	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩)
7	2, 6	erthi 6859	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
8	7	adantrlr 489	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
9	8	adantrrr 491	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
10		simprlr 544	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )
11		simprrr 546	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
12	9, 10, 11	3eqtr4d 2281	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = 𝑞)
13	12	expr 375	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
14	13	exlimdvv 1953	. . . . . . . . 9 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
15	14	exlimdvv 1953	. . . . . . . 8 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
16	15	ex 115	. . . . . . 7 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
17	16	exlimdvv 1953	. . . . . 6 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
18	17	exlimdvv 1953	. . . . 5 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
19	18	impd 254	. . . 4 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
20	19	alrimivv 1928	. . 3 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
21		opeq12 3906	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
22	21	eceq1d 6847	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R )
23	22	eqeq2d 2250	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~R ↔ 𝐴 = [⟨𝑠, 𝑓⟩] ~R ))
24	23	anbi1d 469	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R )))
25		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑤 = 𝑠)
26	25	oveq1d 6103	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·P 𝑢) = (𝑠 ·P 𝑢))
27		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑣 = 𝑓)
28	27	oveq1d 6103	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·P 𝑡) = (𝑓 ·P 𝑡))
29	26, 28	oveq12d 6106	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)))
30	25	oveq1d 6103	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·P 𝑡) = (𝑠 ·P 𝑡))
31	27	oveq1d 6103	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·P 𝑢) = (𝑓 ·P 𝑢))
32	30, 31	oveq12d 6106	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)))
33	29, 32	opeq12d 3912	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩)
34	33	eceq1d 6847	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )
35	34	eqeq2d 2250	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ↔ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ))
36	24, 35	anbi12d 477	. . . . . . 7 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )))
37		opeq12 3906	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ℎ⟩)
38	37	eceq1d 6847	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ℎ⟩] ~R )
39	38	eqeq2d 2250	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~R ↔ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))
40	39	anbi2d 468	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R )))
41		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑢 = 𝑔)
42	41	oveq2d 6104	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·P 𝑢) = (𝑠 ·P 𝑔))
43		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑡 = ℎ)
44	43	oveq2d 6104	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·P 𝑡) = (𝑓 ·P ℎ))
45	42, 44	oveq12d 6106	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)) = ((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)))
46	43	oveq2d 6104	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·P 𝑡) = (𝑠 ·P ℎ))
47	41	oveq2d 6104	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·P 𝑢) = (𝑓 ·P 𝑔))
48	46, 47	oveq12d 6106	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)) = ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔)))
49	45, 48	opeq12d 3912	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩)
50	49	eceq1d 6847	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
51	50	eqeq2d 2250	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ↔ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))
52	40, 51	anbi12d 477	. . . . . . 7 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )))
53	36, 52	cbvex4v 1990	. . . . . 6 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))
54	53	anbi2i 461	. . . . 5 ⊢ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) ↔ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )))
55	54	imbi1i 238	. . . 4 ⊢ (((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
56	55	2albii 1524	. . 3 ⊢ (∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
57	20, 56	sylibr 134	. 2 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
58		eqeq1 2245	. . . . 5 ⊢ (𝑧 = 𝑞 → (𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ↔ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
59	58	anbi2d 468	. . . 4 ⊢ (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
60	59	4exbidv 1923	. . 3 ⊢ (𝑧 = 𝑞 → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
61	60	mo4 2148	. 2 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
62	57, 61	sylibr 134	1 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  ∃wex 1545  ∃*wmo 2087   ∈ wcel 2209  ⟨cop 3712   class class class wbr 4131   × cxp 4774  (class class class)co 6088   Er wer 6808  [cec 6809   / cqs 6810  Pcnp 7664   +_P cpp 7666   ·_P cmp 7667   ~_R cer 7669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-imp 7842  df-enr 8099

This theorem is used by:  mulsrpr  8119